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Theorem sbeqalb 2809
 Description: Theorem *14.121 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 28-Jun-2011.) (Proof shortened by Wolf Lammen, 9-May-2013.)
Assertion
Ref Expression
sbeqalb (A 𝑉 → ((x(φx = A) x(φx = B)) → A = B))
Distinct variable groups:   x,A   x,B
Allowed substitution hints:   φ(x)   𝑉(x)

Proof of Theorem sbeqalb
StepHypRef Expression
1 bibi1 229 . . . . 5 ((φx = A) → ((φx = B) ↔ (x = Ax = B)))
21biimpa 280 . . . 4 (((φx = A) (φx = B)) → (x = Ax = B))
32biimpd 132 . . 3 (((φx = A) (φx = B)) → (x = Ax = B))
43alanimi 1345 . 2 ((x(φx = A) x(φx = B)) → x(x = Ax = B))
5 sbceqal 2808 . 2 (A 𝑉 → (x(x = Ax = B) → A = B))
64, 5syl5 28 1 (A 𝑉 → ((x(φx = A) x(φx = B)) → A = B))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1240   = wceq 1242   ∈ wcel 1390 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-sbc 2759 This theorem is referenced by:  iotaval  4821
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